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duct in question cannot be greater than the sector; for then would it be greater than some circumscribed solid, and therefore (12.) the base of the sector greater than the convex surface of such a solid, which (19.) is impossible: neither can it be less, for then would it be less also than some inscribed solid; which is impossible, because, not only is the base of the sector greater (19.) than the convex surface of such a solid, but the radius C H is likewise greater than the perpendicular CE. Therefore it must be equal to the sector. Therefore, &c.
Every spherical segment, upon a single base, is equal to the half of a cylinder having the same base and the same altitude, together with a sphere, of which that altitude is the diameter.
Let ADFG be any circular halfsegment, by the revolution of which about the diameter A B, a spherical segment having the altitude AG is generated the spherical segment shall be equal to the half of a cylinder having the same base and altitude, together with a sphere, of which A G is the diameter.
Join AF; and from the centre C draw CE perpendicular to A F. Then, because the solid, generated by the segment A D F, is equal to the difference of the solids generated by the sector CADF and the triangle CAF, the former of which (21.) is equal to one.. third of the product of CA by its base AG × 2 × CA (20.), and the latter (12.) to one-third of the product of CE by the convex surface generated by AF viz. (12.) AG × 2 × CE; the solid, generated by the segment AD F, is equal to × AG (Č A2 – CE2), i. e.x AG x AE, or to AG × A F2, because (I. 36. Cor. 1.) CA CE is equal to A E2, and AE2 (III. 3.) to a fourth of A F2. To this add the cone generated by the triangle A FG, which (9.) is equal to × AG × FG2: therefore, the spherical segment in question is equal to T×AG (A F2 + 2 F G2), i. e. to × AG (3 FG2 +AG) because AF2 is equal to AG2+ FG (III.36.). And × AG(3 FG2 + AG) is equal to × AG × FG2 +
× AG3; the first part of which is (4. Cor. 1.) the half of a cylinder having the altitude AG and the same base
with the segment, and the other part (17. Cor. 4.) a sphere of which A G is the diameter. Therefore, &c.
Every double-based spherical segment is equal to the half of a cylinder having the same altitude with the segment and a base equal to the sum of its two bases, together with a sphere of which that altitude is the diameter.
Let FHKG be any portion of the semicircle AD B, by the revolution of which about the diameter A B, a doublebased segment, having the altitude G K, is generated: the segment shall be equal to the half of a cylinder having the same altitude and a base equal to the sum of its two bases, together with a sphere of which G K is the diameter.
For, in the first place, because the solid, generated by the revolution of the circular segment FH,
is equal to the excess of the difference of the sectors generated by CAH and CAF above the solid generated by the triangle CFH, it may be shown by the same steps as in the last proposition, that the solid generated by the revolution of FH is equal to GK FH, i. e. if FL be drawn parallel to GK, to GK (FL2 + LH2). But (I. 22.) FL is equal to G K, and LH is equal to the difference of F G and HK (Î. 22.): therefore (I. 33.), LH is equal to FG2+HK2-2 FGX HK, and × GK (F L2 + LH2), or the solid generated by the segment FH, is equal to 7 × G K (FG2+H K2-2 FG xHK) + ×G K3. To this add the truncated cone generated by the trapezoid FG KH; which (11.) is equal to × GK (FG2 + H K2 + F G × HK): therefore, the double-based segment in question is equal to GK (3 FG2 + 3 H K2) + ¦ ≈ + G K3; or to × GK (F G2 + H K2) + ¦ ø x G K; the first part of which is the half of a cylinder (4. Cor. 1.) having the altitude G K, and a base × FG2 +× H K2, equal to the sum of the bases of the segment, and the other part a sphere of which G K is the diameter (17. Cor. 4.).
Cor. It appears from the demonstrations of this and the preceding proposition, that the solid generated by the re
volution of any circular segment about a diameter of the circle, is equal to
× GK FH2; GK being that portion of the diameter, which is intercepted between two perpendiculars drawn to it from the extremities of the segment, and FH the chord which is the base of the segment. PROP. 24.
Every spherical orb is equal to the sum of three pyramids having their common altitude equal to the thickness of the orb, and for their bases its exterior and interior surfaces, and a mean proportional between them.
For, a spherical orb is the difference between two concentric spheres. Now, if a pyramid be described having its base equal to the exterior surface, or surface of the larger sphere, and
its altitude equal to the radius of that surface, this pyramid will be equal to the whole sphere (IV. 32. Cor. 1. and 17.). And if, from this, there be cut off, by a plane parallel to the base, a pyramid, having its altitude equal to the radius of the interior surface, the two pyramids will be to one another as the cubes of any two homologous edges (IV. 34. Cor.); or, since it may be shown that their altitudes are to one another in the same ratio with the homologous edges, as the cubes of their altitudes, (IV. 27. Cor. 3.), that is, as the cubes of the radii of the spheres, or (18.) as the spheres. Therefore, because the larger pyramid is equal to the larger sphere, the smaller pyramid is equal to the smaller sphere (II. 18.); and the difference of the two pyramids is equal to the difference of the two spheres, that is, the frustum is equal to the spherical orb. And, because the larger base of the frustum is equal to the surface of the larger sphere, it may be shown that its smaller base is equal to the surface of the smaller sphere, exactly in the same manner as it has been already shown, that the content of the smaller pyramid is equal to the content of the smaller sphere; also the altitude of the frustum is equal to the thickness of the orb. But the frustum is equal to the sum of three pyramids, having the same altitude with it, and for their bases its
AB; and from C let CD, CE be drawn in the planes A D B, A E B, perpendicular to AB (I. 44.): the ungula ADBE shall be to the whole sphere, as the angle D CE to four right angles.
For, since the plane DCE is perpendicular to A B (IV.3.), the angle, which measures the inclination of any two planes passing through AB, may be drawn in that plane at the point C (IV. 17. Schol.); and, if any two of these angles at C be equal to one another, the dihedral angles which they measure will be equal (IV.17.), and therefore the ungulas, which have those dihedral angles, may be made to coincide, and are equal to one another. Now, let the angle DCE be divided into any number of equal angles D C F, FCG, &c.; and therefore the dihedral angle DAB E into the same number of dihedral angles by the planes ACF, ACG, &c. (IV. 17.) ; and the ungula ADBE into as many equal ungulas A DBF, ADBG, &c. by the same planes. Then, if the angle DCF be contained in the four right angles about C any number of times exactly, or with a remainder, the ungula ADBF will be contained in the whole sphere the same number of times exactly, or with a remainder. Therefore the ungula ADBE is to the whole sphere as the angle DC E to four right angles (II. def. 7.).
We might here add the proportions of similar segments, sectors, orbs, ungulas, and of their convex surfaces. The reader will, however, easily perceive, from the demonstration of prop. 15, that if similar spherical segments and sectors be defined to be such as are generated by similar circular segments and sectors, their surfaces will be as the squares of the radii, and their contents as the cubes of the radii. And the same may be said of similar spherical orbs, defined to be such that the radii of their exterior and interior surfaces are to one another in the same ratio; and of similar ungulas defined to be such as have their dihedral angles equal to one another.
§ 1. Of great and small circles of the Sphere.-§ 2. Of Spherical Triangles.-3. Of equal Portions of Spherical Surface, and the Measure of solid Angles.-§ 4. Problems.
§1.-Of great and small Circles of the Sphere.
Def. 1. If a sphere is cut by a plane which passes through the centre, the section is called a great circle of the sphere; the radius of such a section being the greatest possible, the same, namely, with the radius of the sphere.
From this definition it is evident that a great circle may be made to pass through any two points in the surface of a sphere; and that, if the two points be not opposite extremities of a diameter, only one great circle can be made to pass through them, for its plane must pass through the centre of the sphere, and only one plane can be made to pass through three points which are not in the same straight line (IV. 1.) But through
the two extremities of a diameter, any number of great circles may be made to pass, for they are in the same straight line with the centre of the sphere (IV. 1. Cor. 4.).
2. If a sphere is cut by a plane which does not pass through the centre, the section is called a small circle of the sphere; the radius of such a section being less than that of the sphere.
A circle, it is plain, may be made to pass through any three points in the sphere's surface; and it will be a great or a small circle, according as its plane passes through the centre of the sphere, or otherwise.
3. The axis of any circle of the sphere is that diameter of the sphere which is perpendicular to the plane of the circle; and the extremities of the axis are called the poles of the circle.
such as have their planes parallel. 4. Parallel circles of a sphere are
the same axis and poles; for a straight It is evident that parallel circles have line which is perpendicular to one of two parallel planes is perpendicular to the other likewise (IV. 11.). It may also be observed that two parallel circles cannot both of them pass through the centre of the sphere, that is, they cannot both be great circles of the sphere.
These four definitions may be illustrated by referring to the figure of prop. 1. in which PAP' is a great circle, ABC a small circle, PO P' the axis,
and P, P' the poles of the circle A B C, and A'B'C', ABC are parallel circles of the sphere.
5. Any portion of the circumference of a great circle is called a spherical
Two points are said to be joined on the surface of the sphere when the spherical arc between them is described; and this arc is called the spherical distance of the two points, in order to distinguish it from their direct distance, which is the straight line which joins them. spherical distance of opposite extremities of a diameter of the sphere is evidently half the circumference of a great circle: but the spherical distance of any other two points is less than a semicircumference, being always the lesser of the two arcs into which they divide the great circle which passes through them.
of the sphere are the spherical arcs 6. The polar distances of any circle which join any point in the circumference with the two poles of the circle.
according to the opening between its containing arcs: thus the angle BAC is greater than the angle D A C by the angle BAD.
Every spherical angle is measured by the plane angle which measures the inclination of the planes of the containing For it is easy to perceive, that if this inclination is the same in any two spherical angles, they may be made to coincide, and therefore are equal to one another. If, therefore, the dihedral angle made by the planes of one spherical angle contain any sub-multiple of the dihedral angle made by the planes of another a certain number of times exactly, or with a remainder, the first spherical angle will contain a like submultiple of the other the same number of times exactly or with a remainder; and, therefore, the spherical angles are to one another (II. def. 7.) as the dihedral angles made by their planes, and have the same measures with them.*
8. When one spherical arc standing upon another makes the adjacent spherical angles equal to one another, each of them is called a spherical right an
gle, and the arc which stands upon the other is said to be perpendicular, or at right angles to it.
The terms acute and obtuse are likewise applied to spherical angles, in the same sense as in Book I. def. 11.
It is evident that a spherical right angle is measured by a rectilineal right angle, a spherical acute angle by a rectili
Hence a spherical angle has been defined by some writers to be identical with the dihedral angle of its planes; while others have extended to it the general definition of the angle in which two curves
In the spherical triangles here considered, it is supposed that each of the sides is less than a semicircumference. For, the greatest spherical distance at which two points can be placed is a semicircumference; and if any arc, as PAP', P be taken equal to a semicircumference, its extremities P, P' will be extremities of
a diameter POP' of the sphere, and therefore the same great will pass through both of them,
and any third point Q on the sphere's surface, so that the arcs Q P and Q P' will be arcs, not of different circles, but of the same circle.
Any three points on the sphere's surface may be assumed for the angles of a spherical triangle (see def. 5.), provided they are not in the same great circle, nor any two of them opposite to one another, that is, opposite extremities of a diameter of the sphere.
10. Two spherical triangles are said to be symmetrical, when the sides of the
one are equal to the sides of the other, each to each, but in a reverse order, as ABC and DEF.
11. If ABC is any spherical triangle, and the points A', B', C' are those poles of the arcs B C, A C, A B, respectively, which lie upon the same sides of them with the opposite an
cut one another, considering it the same with the gles A, B, C, and the triangle A'B'C'
plane rectilineal angle of the tangents at the point A; for the latter angle, being contained by perpendiculars to the common section OA, measures the dihedral angle of the planes.
is completed: this triangle A' B' C' is said to be the polar triangle of the triangle A B C.
There are no fewer than eight different triangles which have for their angular points poles of the sides of a given triangle ABC; but there is only one triangle in which these poles A', B', C', lie towards the same parts with the opposite angles A, B, C, and this is the triangle A'B' C', which is known under the name of the polar triangle.
12. A spherical polygon is any portion of the sphere's surface included B by more than three arcs of different great circles, as ABCDE.
one another; because the right-angled triangles OKA, OK B have their hypotenuses OA, OB each a radius of the sphere, and the side OK common to both (I.13.). Therefore, in this case the section is a circle having the centre K. Therefore, &c.
Cor. 1. The radius of a great circle is the same with the radius of the sphere; and the radius-square of a small circle is less than the radius-square of the Esphere by the square of the perpendicular, which is drawn to its plane from the centre of the sphere (I. 36. Cor. 1.). Cor. 2. Every diameter of a great circle is likewise a diameter of the sphere.
13. Opposite points on the surface of the sphere are those which are opposite extremities of a diameter of the sphere. It is evident that the arcs which join two such points with any third point on the sphere's surface, are parts of the same great circle, and are together equal to a semicircumference (see the second figure of def. 9.)
Every plane section of a sphere is a circle; the centre of which is either the centre of the sphere, or the foot of the perpendicular which is drawn to the plane from the centre of the sphere.
The substance of this proposition has been already given in the corollary to Book IV. Prop. 8; and the following demonstration is only a statement at greater length of the reasoning from which it was there inferred.
If the plane pass through the centre O of the sphere, as PAP', the
Either pole of a circle of the sphere is equally distant from all points in the circumference of that circle; whether the direct or the spherical distance be understood.
Let A B C (see the figure of prop. 1.) be any circle of a sphere which has the centre O, and let OK be drawn perpendicular to the plane ABC, and produced to meet the surface of the sphere in P; then, if A, B be any two points in the circumference of the circle ABC, and if the straight lines PA, PB, as also the spherical arcs PA, PB be drawn, the line P A shall be equal to the line PB, and the arc PA to the arc PB.
Join KA, K B. Then, because K is the centre of the circle ABC (1.), the right-angled triangles PK A and PK B have the two sides P K, KA of the one equal to the two sides PK, KB of the other, each to each; therefore, (I. 4.) the hypotenuse PA is equal to the hypotenuse PB. And because, in equal circles, the arcs which are subtended by equal chords are equal to one another (III. 12. Cor. 1.), the arc PA is likewise equal to the arc PB. And in like manner it may be shown that the other pole P' is also equidistant from A and B.
In this demonstration it is supposed that the point K does not coincide with the point O, or that the circle in question is not a great circle. If, however, ABC is a great circle, the angles POA, PO B are right angles, and therefore equal to one another (I. 1.), from which the equality of the chords PA, PB and of the arcs PA, PB will follow as before.
Cor. 1. Hence any circle of a sphere may be conceived to be described from